Why Jumps Can Separate Variance Swap Market and Replication Prices

# Market price versus theoretical price of varswaps


# Market price versus theoretical price of varswaps












When I traded varswaps several years ago, for some indices there was a significant mismatch between market price and theoretical price.

The theoretical price assumes continuous monitoring and infinite strip of options as replicating portfolio.

The market price, as specificed by the term sheet, is based on among others discrete monitoring (e.g. daily).

I am wondering what the biggest drivers are of the mismatch between theoretical price and market price of varswaps.

Is it the discrete versus continuous monitoring, the impossibility of replicating with infinite strip of options, supply and demand, or all of the above? Is there a way to break down the difference into these components? Are there papers that have discussed this?

EDIT:

My current thinking about this, which I'd like to write down in a note at some point perhaps, is as follows:

As mentioned by Quantuple below, the term sheet does not say anything about the underlying model. All it it says, in the discrete monitoring case, is that the payoff is determined by the following formula:

$$ \sum_{i=0}^{N-1} \left( \frac{S_{i+1}}{S_i} - 1 \right)^2 - L^2 $$

with $L^2$ the strike of the contract.

Given forward start options $$ C(K,t,i,i+1) = E_t \left[ \left(\frac{S_{i+1}}{S_i} - K \right)_+ \right] $$ the payoff of the discretely monitored variance swap can be replicated (using Carr-Madan) regardless of underlying dynamics. Using discrete log-returns squared does not change the argument I think of having to use forward starts for proper and model-independent replication.

My hypotheses/idea is thus that the premium between discretely and continuously monitored varswap is or should be actually the difference between replication using vanillas (and thus model error from among others the diffusion assumption) versus replication using forward starts that as far as I can see would replicate it model-independently (jumps included).

Would this make sense?

## Answer by Quantuple (score 6, accepted)

https://quant.stackexchange.com/a/63880

The well-known formula expressing the price of a variance swap as a function of an (infinite) strip of European options is actually not that model-free: it assumes that the price process follows a pure diffusion, i.e. that it does not jump.

So my two cents would be:

- Even if you could trade an infinite strip of options you would only be able to replicate the quantity $\sigma^2_{cont\_nojump}$ (integrated quadratic variation of a pure diffusion process)

- As you point out, $\sigma^2_{cont\_nojump}$ is the continuous time limit of the actual contract specification $\sigma^2_{discr\_nojump}$ which is discrete sampled.

- Furthermore the actual contract specification is purely descriptive: it makes no underlying assumptions about jumps. As such, determining the price of the contract by using a formula (hence setting up a replication strategy) which only prevails in the absence of jumps might play tricks on you (especially if we look at event-driven markets like single names' options)

So I would tend to say that the effects matter in that order: jumps (offer-demand/events) > discrete sampling > perfect replication.

For references, I would recommend "The Effects of Jumps and Discrete Sampling on Volatility and Variance Swaps" by Broadie et al, where they indicate in the abstract that

> For realistic contract specifications and model parameters, we find that the effect of discrete sampling is typically small while the effect of jumps can be significant.

Similarly, Elie Ayache has an interesting discussion on the Variance Swaps market in a paper called "The Irony in the Variance Swaps".

Naturally you'll find further useful references in each of these papers.